theory of ferromagnetic unconventional superconductors ......theory of ferromagnetic unconventional...

Chapter 0

Theory of Ferromagnetic UnconventionalSuperconductors with Spin-Triplet Electron Pairing

Dimo I. Uzunov

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48579

1. Introduction

In the beginning of this century the unconventional superconductivity of spin-triplettype had been experimentally discovered in several itinerant ferromagnets. Since thenmuch experimental and theoretical research on the properties of these systems hasbeen accomplished. Here we review the phenomenological theory of ferromagneticunconventional superconductors with spin-triplet Cooper pairing of electrons. Sometheoretical aspects of the description of the phases and the phase transitions inthese interesting systems, including the remarkable phenomenon of coexistence ofsuperconductivity and ferromagnetism are discussed with an emphasis on the comparisonof theoretical results with experimental data.

The spin-triplet or p-wave pairing allows parallel spin orientation of the fermion Cooper pairsin superfluid 3He and unconventional superconductors [1]. For this reason the resultingunconventional superconductivity is robust with respect to effects of external magneticfield and spontaneous ferromagnetic ordering, so it may coexist with the latter. Thisgeneral argument implies that there could be metallic compounds and alloys, for which thecoexistence of spin-triplet superconductivity and ferromagnetism may be observed.

Particularly, both superconductivity and itinerant ferromagnetic orders can be created by thesame band electrons in the metal, which means that spin-1 electron Cooper pairs participatein the formation of the itinerant ferromagnetic order. Moreover, under certain conditions thesuperconductivity is enhanced rather than depressed by the uniform ferromagnetic order thatcan generate it, even in cases when the superconductivity does not appear in a pure form as anet result of indirect electron-electron coupling.

The coexistence of superconductivity and ferromagnetism as a result of collective behaviorof f -band electrons has been found experimentally for some Uranium-based intermetalliccompounds as, UGe2 [2–5], URhGe [6–8], UCoGe [9, 10], and UIr [11, 12]. At low temperature(T ∼ 1 K) all these compounds exhibit thermodynamically stable phase of coexistence ofspin-triplet superconductivity and itinerant ( f -band) electron ferromagnetism (in short, FS

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Chapter 17

2 Will-be-set-by-IN-TECH

phase). In UGe2 and UIr the FS phase appears at high pressure (P ∼ 1 GPa) whereas inURhGe and UCoGe, the coexistence phase persists up to ambient pressure (105Pa ≡ 1bar).

Experiments, carried out in ZrZn2 [13], also indicated the appearance of FS phase at T <1 K in a wide range of pressures (0 < P ∼ 21 kbar). In Zr-based compounds theferromagnetism and the p-wave superconductivity occur as a result of the collective behaviorof the d-band electrons. Later experimental results [14, 15] had imposed the conclusionthat bulk superconductivity is lacking in ZrZn2, but the occurrence of a surface FS phase atsurfaces with higher Zr content than that in ZrZn2 has been reliably demonstrated. Thusthe problem for the coexistence of bulk superconductivity with ferromagnetism in ZrZn2is still unresolved. This raises the question whether the FS phase in ZrZn2 should bestudied by surface thermodynamics methods or should it be investigated by consideringthat bulk and surface thermodynamic phenomena can be treated on the same footing.Taking into account the mentioned experimental results for ZrZn2 and their interpretationby the experimentalists [13–15] we assume that the unified thermodynamic approach can beapplied. As an argument supporting this point of view let us mention that the spin-tripletsuperconductivity occurs not only in bulk materials but also in quasi-two-dimensional (2D)systems – thin films and surfaces and quasi-1D wires (see, e.g., Refs. [16]). In ZrZn2 and UGe2both ferromagnetic and superconducting orders vanish at the same critical pressure Pc, a factimplying that the respective order parameter fields strongly depend on each other and shouldbe studied on the same thermodynamic basis [17].

Fig. 1 illustrates the shape of the T − P phase diagrams of real intermetallic compounds.The phase transition from the normal (N) to the ferromagnetic phase (FM) (in short, N-FMtransition) is shown by the line TF(P). The line TFS(P) of the phase transition from FM toFS (FM-FS transition) may have two or more distinct shapes. Beginning from the maximal(critical) pressure Pc, this line may extend, like in ZrZn2, to all pressures P < Pc, includingthe ambient pressure Pa; see the almost straight line containing the point 3 in Fig. 1. A secondpossible form of this line, as known, for example, from UGe2 experiments, is shown in Fig. 1by the curve which begins at P ∼ Pc, passes through the point 2, and terminates at somepressure P1 > Pa, where the superconductivity vanishes. These are two qualitatively differentphysical pictures: (a) when the superconductivity survives up to ambient pressure (type I),and (b) when the superconducting states are possible only at relatively high pressure (forUGe2, P1 ∼ 1 GPa); type II. At the tricritical points 1, 2 and 3 the order of the phase transitionschanges from second order (solid lines) to first order (dashed lines). It should be emphasized

that in all compounds, mentioned above, TFS(P) is much lower than TF(P) when the pressureP is considerably below the critical pressure Pc (for experimental data, see Sec. 8).

In Fig. 1, the circle C denotes a narrow domain around Pc at relatively low temperatures(T � 300 mK), where the experimental data are quite few and the predictions about theshape of the phase transition are not reliable. It could be assumed, as in the most part of theexperimental papers, that (T = 0, P = Pc) is the zero temperature point at which both linesTF(P) and TFS(P) terminate. A second possibility is that these lines may join in a single (N-FS)phase transition line at some point (T � 0, P′

c � Pc) above the absolute zero. In this secondvariant, a direct N-FS phase transition occurs, although this option exists in a very smalldomain of temperature and pressure variations: from point (0, Pc) to point (T � 0, P′

c � Pc).A third variant is related with the possible splitting of the point (0, Pc), so that the N-FM lineterminates at (0, Pc), whereas the FM-FS line terminates at another zero temperature point

416 Superconductors – Materials, Properties and Applications

Theory of Ferromagnetic Unconventional Superconductors with Spin-Triplet Electron Pairing 3

P

T

Pc

FM

N

FS 3

TF(P)

1

2

C

type I type II

P1

Figure 1. An illustration of T − P phase diagram of p-wave ferromagnetic superconductors (details areomitted): N – normal phase, FM – ferromagnetic phase, FS – phase of coexistence of ferromagnetic orderand superconductivity, TF(P) and TFS(P) are the respective phase transition lines: solid lines correspondto second order phase transitions, dashed lines stand for first order phase transition; 1 and 2 aretricritical points; Pc is the critical pressure, and the circle C surrounds a relatively small domain of highpressure and low temperature, where the phase diagram may have several forms depending on theparticular substance. The line of the FM-FS phase transition may extend up to ambient pressure (type Iferromagnetic superconductors), or, may terminate at T = 0 at some high pressure P = P1 (type IIferromagnetic superconductors, as indicated in the figure).

(0, P0c); P0c � Pc. In this case, the p-wave ferromagnetic superconductor has three points ofquantum (zero temperature) phase transitions [18, 19].

These and other possible shapes of T − P phase diagrams are described within the frameworkof the general theory of Ginzburg-Landau (GL) type [18–20] in a conformity with theexperimental data; see also Ref. [21]. The same theory has been confirmed by a microscopicderivation based on a microscopic Hamiltonian including a spin-generalized BCS term andan additional Heisenberg exchange term [22].

For all compounds, cited above, the FS phase occurs only in the ferromagnetic phase domainof the T − P diagram. Particularly at equilibrium, and for given P, the temperature TF(P) ofthe normal-to-ferromagnetic phase (or N-FM) transition is never lower than the temperatureTFS(P) of the ferromagnetic-to-FS phase transition (FM-FS transition). This confirms thepoint of view that the superconductivity in these compounds is triggered by the spontaneousmagnetization M, in analogy with the well-known triggering of the superfluid phase A1in 3He at mK temperatures by the external magnetic field H. Such “helium analogy" hasbeen used in some theoretical studies (see, e.g., Ref. [23, 24]), where Ginzburg-Landau (GL)free energy terms, describing the FS phase were derived by symmetry group arguments.The non-unitary state, with a non-zero value of the Cooper pair magnetic moment, knownfrom the theory of unconventional superconductors and superfluidity in 3He [1], has beensuggested firstly in Ref. [23], and later confirmed in other studies [7, 24]; recently, the sametopic was comprehensively discussed in Ref. [25].

417Theory of Ferromagnetic Unconventional Superconductors with Spin-Triplet Electron Pairing


For the spin-triplet ferromagnetic superconductors the trigger mechanism was recentlyexamined in detail [20, 21]. The system main properties are specified by terms in the GLexpansion of form Miψjψk, which represent the interaction of the magnetization M = {Mj; j =1, 2, 3} with the complex superconducting vector field ψ = {ψj; j = 1, 2, 3}. Particularly, theseterms are responsible for the appearance of superconductivity (|ψ| > 0) for certain T and Pvalues. A similar trigger mechanism is familiar in the context of improper ferroelectrics [26].

A crucial feature of these systems is the nonzero magnetic moment of the spin-triplet Cooperpairs. As mentioned above, the microscopic theory of magnetism and superconductivityin non-Fermi liquids of strongly interacting heavy electrons ( f and d band electrons) iseither too complex or insufficiently developed to describe the complicated behavior initinerant ferromagnetic compounds. Several authors (see [20, 21, 23–25]) have exploredthe phenomenological description by a self-consistent mean field theory, and here we willessentially use the thermodynamic results, in particular, results from the analysis in Refs. [20,21]. Mean-field microscopic theory of spin-mediated pairing leading to the mentionednon-unitary superconductivity state has been developed in Ref. [17] that is in conformity withthe phenomenological description that we have done.

The coexistence of s-wave (conventional) superconductivity and ferromagnetic order is along-standing problem in condensed matter physics [27–29]. While the s-state Cooper pairscontain only opposite electron spins and can easily be destroyed by the spontaneous magneticmoment, the spin-triplet Cooper pairs possess quantum states with parallel orientationof the electron spins and therefore can survive in the presence of substantial magneticmoments. This is the basic difference in the magnetic behavior of conventional (s-state) andspin-triplet superconductivity phases. In contrast to other superconducting materials, forexample, ternaty and Chevrel phase compounds, where the effect of magnetic order on s-wavesuperconductivity has been intensively studied in the seventies and eighties of last century(see, e.g., Refs. [27–29]), in these ferromagnetic compounds the phase transition temperatureTF to the ferromagnetic state is much higher than the phase transition temperature TFS fromferromagnetic to a (mixed) state of coexistence of ferromagnetism and superconductivity. Forexample, in UGe2 we have TFS ∼ 0.8 K versus maximal TF = 52 K [2–5]. Another importantdifference between the ternary rare earth compounds and the intermetallic compounds (UGe2,UCoGe, etc.), which are of interest in this paper, is that the experiments with the latter do notgive any evidence for the existence of a standard normal-to-superconducting phase transitionin zero external magnetic field. This is an indication that the (generic) critical temperature Tsof the pure superconductivity state in these intermetallic compounds is very low (Ts � TFS),if not zero or even negative.

In the reminder of this paper, we present general thermodynamic treatment of systemswith itinerant ferromagnetic order and superconductivity due to spin-triplet Cooper pairingof the same band electrons, which are responsible for the spontaneous magnetic moment.The usual Ginzburg-Landau (GL) theory of superconductors has been completed toinclude the complexity of the vector order parameter ψ, the magnetization M and newrelevant energy terms [20, 21]. We outline the T − P phase diagrams of ferromagneticspin-triplet superconductors and demonstrate that in these materials two contrasting typesof thermodynamic behavior are possible. The present phenomenological approach includesboth mean-field and spin-fluctuation theory (SFT), as the arguments in Ref. [30]. We propose



a simple, yet comprehensive, modeling of P dependence of the free energy parameters,resulting in a very good compliance of our theoretical predictions for the shape the T − Pphase diagrams with the experimental data (for some preliminary results, see Ref. [18, 19]).

The theoretical analysis is done by the standard methods of phase transition theory [31].Treatment of fluctuation effects and quantum correlations [31, 32] is not included in this study.But the parameters of the generalized GL free energy may be considered either in mean-fieldapproximation as here, or as phenomenologically renormalized parameters which are affectedby additional physical phenomena, as for example, spin fluctuations.

We demonstrate with the help of present theory that we can outline different possibletopologies for the T − P phase diagram, depending on the values of Landau parameters,derived from the existing experimental data. We show that for spin-triplet ferromagneticsuperconductors there exist two distinct types of behavior, which we denote as Zr-type(or, alternatively, type I) and U-type (or, type II); see Fig. 1. This classification of the FS,first mentioned in Ref. [18], is based on the reliable interrelationship between a quantitativecriterion derived by us and the thermodynamic properties of the ferromagnetic spin-tripletsuperconductors. Our approach can be also applied to URhGe, UCoGe, and UIr. Theresults shed light on the problems connected with the order of the quantum phase transitionsat ultra-low and zero temperatures. They also raise the question for further experimentalinvestigations of the detailed structure of the phase diagrams in the high-P/low-T region.

2. Theoretical framework

Consider the GL free energy functional of the form

F(ψ, B) =∫

Vdx

[fS(ψ) + fF(M) + fI(ψ, M) +

B2

8π− B.M

], (1)

where the fields ψ, M, and B are supposed to depend on the spatial vector x ∈ V in thevolume V of the superconductor. In Eq. (1), the free energy density generated by the genericsuperconducting subsystem (ψ) is given by

fS(ψ) = fgrad(ψ) + as|ψ|2 + bs

2|ψ|4 + us

2|ψ2|2 + vs

2

3

∑j=1

|ψj|4 , (2)

with

fgrad(ψ) = K1(Diψj)∗(DiDj) + K2

[(Diψi)

∗(Djψj) + (Diψj)∗(Djψi)

](3)

+K3(Diψi)∗(Diψi),

where a summation over the indices (i, j) is assumed, the symbol Dj = (h∂/i∂xj + 2|e|Aj/c)of covariant differentiation is introduced, and Kj are material parameters [1]. The free energydensity fF(M) of a standard ferromagnetic phase transition of second order [31] is

fF(M) = c f

3

∑j=1

|∇M j|2 + a f M2 +b f

2M4, (4)



with c f , b f > 0, and a f = α(T − Tf ), where α f > 0 and Tf is the critical temperature,corresponding of the generic ferromagnetic phase transition. Finally, the energy fI(ψ, M)produced by the possible couplings of ψ and M is given by

fI(ψ, M) = iγ0M.(ψ × ψ∗) + δ0 M2|ψ|2, (5)

where the coupling parameter γ0 ∼ J depends on the ferromagnetic exchange parameterJ > 0, [23, 24] and δ0 is the standard M − ψ coupling parameter, known from the theoryof multicritical phenomena [31] and from studies of coexistence of ferromagnetism andsuperconductivity in ternary compounds [27, 28].

As usual, in Eq. (2), as = (T − Ts), where Ts is the critical temperature of the genericsuperconducting transition, bs > 0. The parameters us and vs and δ0 may take some negativevalues, provided the overall stability of the system is preserved. The values of the materialparameters μ = (Ts, Tf , αs, α f , bs, us, vs, b f , Kj, γ0 and δ0) depend on the choice of the substanceand on intensive thermodynamic parameters, such as the temperature T and the pressure P.From a microscopic point of view, the parameters μ depend on the density of states UF(kF)on the Fermi surface. On the other hand UF varies with T and P. Thus the relationships(T, P) � UF � μ, i.e., the functional relations μ[UF(T, P)], are of essential interest. Whilethese relations are unknown, one may suppose some direct dependence μ(T, P). The lattershould correspond to the experimental data.

The free energy (1) is quite general. It has been deduced by several reliable arguments. Inorder to construct Eq. (1)–(5) we have used the standard GL theory of superconductors andthe phase transition theory with an account of the relevant anisotropy of the p-wave Cooperpairs and the crystal anisotropy, described by the us- and vs-terms in Eq. (2), respectively.Besides, we have used the general case of cubic anisotropy, when all three components ψj ofψ are relevant. Note, that in certain real cases, for example, in UGe2, the crystal symmetry istetragonal, ψ effectively behaves as a two-component vector and this leads to a considerablesimplification of the theory. As shown in Ref. [20], the mentioned anisotropy terms are notessential in the description of the main thermodynamic properties, including the shape of theT − P phase diagram. For this reason we shall often ignore the respective terms in Eq. (2).The γ0-term triggers the superconductivity (M-triger effect [20, 21]) while the δ0M2|ψ|2–termmakes the model more realistic for large values of M. This allows for an extension of thedomain of the stable ferromagnetic order up to zero temperatures for a wide range of valuesof the material parameters and the pressure P. Such a picture corresponds to the real situationin ferromagnetic compounds [20].

The total free energy (1) is difficult for a theoretical investigation. The various vortex anduniform phases described by this complex model cannot be investigated within a singlecalculation but rather one should focus on particular problems. In Ref. [24] the vortex phasewas discussed with the help of the criterion [33] for a stability of this state near the phasetransition line Tc2(B), ; see also, Ref. [34]. The phase transition line Tc2(H) of a usualsuperconductor in external magnetic field H = |H| is located above the phase transition lineTs of the uniform (Meissner) phase. The reason is that Ts is defined by the equation as(T) = 0,whereas Tc2(H) is a solution of the equation |as| = μB H, where μB = |e|h/2mc is theBohr magneton [34]. For ferromagnetic superconductors, where M > 0, one should use themagnetic induction B rather than H. In case of H = 0 one should apply the same criterion withrespect to the magnetization M for small values of |ψ| near the phase transition line Tc2(M);



M = |M|. For this reason we should use the diagonal quadratic form [35] correspondingto the entire ψ2-part of the total free energy functional (1). The lowest energy term in thisdiagonal quadratic part contains a coefficient a of the form a = (as − γ0M − δM2) [35]. Nowthe equation a(T) = 0 defines the critical temperature of the Meissner phase and the equation|as| = μBM stands for Tc2(M). It is readily seen that these two equations can be writtenin the same form, provided the parameter γ0 in a is substituted by γ′

0 = (γ0 − μB). Thusthe phase transition line corresponding to the vortex phase, described by the model (1) atzero external magnetic field and generated by the magnetization M, can be obtained fromthe phase transition line corresponding to the uniform superconducting phase by an effectivechange of the value of the parameter γ0. Both lines have the same shape and this is a particularproperty of the present model. The variation of the parameter γ0 generates a family of lines.

Now we propose a possible way of theoretical treatment of the TFS(P) line of the FM-FS phasetransition, shown in Fig. (1). This is a crucial point in our theory. The phase transition line ofthe uniform superconducting phase can be calculated within the thermodynamic analysis ofthe uniform phases, described by the free energy (1). This analysis is done in a simple variantof the free energy (1) in which the fields ψ and M do not depend on the spatial vector x.The accomplishment of such analysis will give a formula for the phase transition line TFS(P)which corresponds a Meissner phase coexisting with the ferromagnetic order. The theoreticalresult for TFS(P) will contain a unspecified parameter γ0. If the theoretical line TFS(P) isfitted to the experimental data for the FM-FS transition line corresponding to a particularcompound, the two curves will coincide for some value of γ0, irrespectively on the structureof the FS phase. If the FS phase contains a vortex superconductivity the fitting parameterγ0(e f f ) should be interpreted as γ′

0 but if the FS phase contains Meissner superconductivity,γ0(e f f ) should be identified as γ0. These arguments justify our approach to the investigationof the experimental data for the phase diagrams of intermetallic compounds with FM and FSphases. In the remainder of this paper, we shall investigate uniform phases.

3. Model considerations

In the previous section we have justified a thermodynamic analysis of the free energy (1) interms of uniform order parameters. Neglecting the x-dependence of ψ and M, the free energyper unit volume, F/V = f (ψ, M) in zero external magnetic field (H = 0), can be written inthe form

f (ψ, M) = as|ψ|2 + bs

2|ψ|4 + us

2|ψ2|2 + vs

2

3

∑j=1

|ψj|4 + a f M2 +b f

2M4 (6)

+ iγ0M · (ψ × ψ∗) + δ0 M2|ψ|2.

Here we slightly modify the parameter a f by choosing a f = α f [Tn − Tnf (P)], where n =

1 gives the standard form of a f , and n = 2 applies for SFT [30] and the Stoner-Wohlfarthmodel [36]. Previous studies [20] have shown that the anisotropy represented by the us andvs terms in Eq. (6) slightly perturbs the size and shape of the stability domains of the phases,while similar effects can be achieved by varying the bs factor in the bs|ψ|4 term. For thesereasons, in the present analysis we ignore the anisotropy terms, setting us = vs = 0, andconsider bs ≡ b > 0 as an effective parameter. Then, without loss of generality, we are free tochoose the magnetization vector to have the form M = (0, 0, M).



According to the microscopic theory of band magnetism and superconductivity themacroscopic material parameters in Eq. (6) depend in a quite complex way on the densityof states at the Fermi level and related microscopic quantities [37]. That is why we canhardly use the microscopic characteristics of these complex metallic compounds in order toelucidate their thermodynamic properties, in particular, in outlining their phase diagrams insome details. However, some microscopic simple microscopic models reveal useful results,for example, the zero temperature Stoner-type model employed in Ref. [38].

We redefine for convenience the free energy (6) in a dimensionless form by f = f /(b f M40),

where M0 = [α f Tnf 0/b f ]

1/2 > 0 is the value of the magnetization M corresponding to the puremagnetic subsystem (ψ ≡ 0) at T = P = 0 and Tf 0 = Tf (0). The order parameters assumethe scaling m = M/M0 and ϕ = ψ/[(b f /b)1/4M0], and as a result, the free energy becomes

f = rφ2 +φ4

2+ tm2 +

m4

2+ 2γmφ1φ2sinθ + γ1m2φ2, (7)

where φj = |ϕj|, φ = |ϕ|, and θ = (θ2 − θ1) is the phase angle between the complex ϕ1 =

φ1eiθ1 and ϕ2 = φ2eθ2 . Note that the phase angle θ3, corresponding to the third complex fieldcomponent ϕ3 = φ3eiθ3 does not enter explicitly in the free energy f , given by Eq. (7), whichis a natural result of the continuous space degeneration. The dimensionless parameters t, r, γand γ1 in Eq. (7) are given by

t = Tn − Tnf (P), r = κ(T − Ts), (8)

where κ = αsb1/2f /α f b1/2Tn−1

f 0 , γ = γ0/[α f Tnf 0b]1/2, and γ1 = δ0/(bb f )

1/2. The reduced

temperatures are T = T/Tf 0, Tf (P) = Tf (P)/Tf 0, and Ts(P) = Ts(P)/Tf 0.

The analysis involves making simple assumptions for the P dependence of the t, r, γ, and γ1parameters in Eq. (7). Specifically, we assume that only Tf has a significant P dependence,described by

Tf (P) = (1 − P)1/n, (9)

where P = P/P0 and P0 is a characteristic pressure deduced later. In ZrZn2 and UGe2the P0 values are very close to the critical pressure Pc at which both the ferromagnetic andsuperconducting orders vanish, but in other systems this is not necessarily the case. As wewill discuss, the nonlinearity (n = 2) of Tf (P) in ZrZn2 and UGe2 is relevant at relatively highP, at which the N-FM transition temperature TF(P) may not coincide with Tf (P); TF(P) isthe actual line of the N-FM phase transition, as shown in Fig. (1). The form (9) of the modelfunction Tf (P) is consistent with preceding experimental and theoretical investigations of theN-FM phase transition in ZrZn2 and UGe2 (see, e.g., Refs. [4, 24, 39]). Here we consider onlynon-negative values of the pressure P (for effects at P < 0, see, e.g., Ref. [44]).

The model function (9) is defined for P ≤ P0, in particular, for the case of n > 1, but we shouldhave in mind that, in fact, the thermodynamic analysis of Eq. (7) includes the parameter trather than Tf (P). This parameter is given by

t(T, P) = Tn − 1 + P, (10)



and is well defined for any P. This allows for the consideration of pressures P > P0 within thefree energy (7).

The model function Tf (P) can be naturally generalized to Tf (P) = (1− Pβ)1/α but the presentneeds of interpretation of experimental data do not require such a complex consideration(hereafter we use Eq. (9) which corresponds to β = 1 and α = n). Besides, other analyticalforms of Tf (P) can also be tested in the free energy (7), in particular, expansion in powers ofP, or, alternatively, in (1 − P) which satisfy the conditions Tf (0) = 1 and Tf (1) = 0. Note,that in URhGe the slope of TF(P) ∼ Tf (P) is positive from P = 0 up to high pressures [8] andfor this compound the form (9) of Tf (P) is inconvenient. Here we apply the simplest variantsof P-dependence, namely, Eqs. (9) and (10).

In more general terms, all material parameters (r, t, γ, . . . ) may depend on the pressure. Wesuppose that a suitable choice of the dependence of t on P is enough for describing the mainthermodynamic properties and this supposition is supported by the final results, presented inthe remainder of this paper. But in some particular investigations one may need to introducea suitable pressure dependence of other parameters.

4. Stable phases

The simplified model (7) is capable of describing the main thermodynamic properties ofspin-triplet ferromagnetic superconductors. For r > 0, i.e., T > Ts, there are three stablephases [20]: (i) the normal (N-) phase, given by φ = m = 0 (stability conditions: t ≥ 0,r ≥ 0); (ii) the pure ferromagnetic phase (FM phase), given by m = (−t)1/2 > 0, φ = 0,which exists for t < 0 and is stable provided r ≥ 0 and r ≥ (γ1t + γ|t|1/2), and (iii) thealready mentioned phase of coexistence of ferromagnetic order and superconductivity (FSphase), given by sinθ = ∓1, φ3 = 0, φ1 = φ2 = φ/

√2, where

φ2 = κ(Ts − T)± γm − γ1m2 ≥ 0. (11)

The magnetization m satisfies the equation

c3m3 ± c2m2 + c1m ± c0 = 0 (12)

with coefficients c0 = γκ(T − Ts),

c1 = 2[

Tn + κγ1(Ts − T) + P − 1 − γ2

2

], (13)

c2 = 3γγ1, c3 = 2(1 − γ21). (14)

The FS phase contains two thermodynamically equivalent phase domains that can bedistinguished by the upper and lower signs (±) of some terms in Eqs. (11) and (12). Theupper sign describes the domain (labelled bellow again by FS), where m > 0, sinθ = −1,whereas the lower sign describes the conjunct domain FS∗, where m < 0 and sinθ = 1(for details, see, Ref. [20]). Here we consider one of the two thermodynamically equivalentphase domains, namely, the domain FS, which is stable for m > 0 (FS∗ is stable for m < 0).This “one-domain approximation" correctly presents the main thermodynamic properties



N TN tN PN(n)A Ts γ2/2 1 − Tn

s + γ2/2B Ts + γ2(2 + γ1)/4κ(1 + γ1)

2 −γ2/4(1 + γ1)2 1 − Tn

B − γ2/4(1 + γ1)2

C Ts + γ2/4κ(1 + γ1) 0 1 − TnC

max Ts + γ2/4κγ1 −γ2/4γ21 1 − Tn

m − γ2/4γ21

Table 1. Theoretical results for the location [(T, P) - reduced coordinates] of the tricritical points A≡ (TA, PA) and B ≡ (TB, PB), the critical-end point C ≡ (TC, PC), and the point of temperature maximum,max =(Tm, Pm) on the curve TFS(P) of the FM-FS phase transitions of first and second orders (for details,see Sec. 5). The first column shows TN ≡ T(A,B,C,m). The second column stands for tN = t(A,B,C,m). Thereduced pressure values P(A,B,C,m) of points A, B, C, and max are denoted by PN(n): n = 1 stands for thelinear dependence Tf (P), and n = 2 stands for the nonlinear Tf (P) and t(T), corresponding to SFT.

described by the model (6), in particular, in the case of a lack of external symmetry breakingfields. The stability conditions for the FS phase domain given by Eqs.(11) and (12) are γM ≥ 0,

κ(Ts − T)± γm − 2γ1m2 ≥ 0, (15)

and

3(1 − γ21)m

2 + 3γγ1m + Tn − 1 + P + κγ1(Ts − T)− γ2

2≥ 0. (16)

These results are valid whenever Tf (P) > Ts(P), which excludes any pure superconductingphase (ψ = 0, m = 0) in accord with the available experimental data.

For r < 0, and t > 0 the models (6) and (7) exhibit a stable pure superconducting phase (φ1 =φ2 = m = 0, φ2

3 = −r) [20]. This phase may occur in the temperature domain Tf (P) < T < Ts.For systems, where Tf (0) � Ts, this is a domain of pressure in a very close vicinity of P0 ∼ Pc,where TF(P) ∼ Tf (P) decreases up to values lower than Ts. Of course, such a situation isdescribed by the model (7) only if Ts > 0. This case is interesting from the experimentalpoint of view only when Ts > 0 is enough above zero to enter in the scope of experimentallymeasurable temperatures. Up to date a pure superconducting phase has not been observedwithin the accuracy of experiments on the mentioned metallic compounds. For this reason, inthe reminder of this paper we shall often assume that the critical temperature Ts of the genericsuperconducting phase transition is either non-positive (Ts ≤ 0), or, has a small positive valuewhich can be neglected in the analysis of the available experimental data.

The negative values of the critical temperature Ts of the generic superconducting phasetransition are generally possible and produce a variety of phase diagram topologies (Sec. 5).Note, that the value of Ts depends on the strength of the interaction mediating the formationof the spin-triplet Cooper pairs of electrons. Therefore, for the sensitiveness of such electroncouplings to the crystal lattice properties, the generic critical temperature Ts depends on thepressure. This is an effect which might be included in our theoretical scheme by introducingsome convenient temperature dependence of Ts. To do this we need information either fromexperimental data or from a comprehensive microscopic theory.

Usually, Ts ≤ 0 is interpreted as a lack of any superconductivity but here the samenon-positive values of Ts are effectively enhanced to positive values by the interactionparameter γ which triggers the superconductivity up to superconducting phase-transitiontemperatures TFS(P) > 0. This is readily seen from Table 1, where we present the reducedcritical temperatures on the FM-FS phase transition line TFS(P), calculated from the present



theory, namely, Tm – the maximum of the curve TFS(P) (if available, see Sec. 5), thetemperatures TA and TB, corresponding to the tricritical points A ≡ (TA, PA) and B ≡ (TB, PB),and the temperature TC, corresponding to the critical-end point C ≡ (TC, PC). The theoreticalderivation of the dependence of the multicritical temperatures TA, TB and TC on γ, γ1, κ, andTs, as well as the dependence of Tm on the same model parameters is outlined in Sec. 5. Allthese temperatures as well as the whole phase transition line TFS(P) are considerably boostedabove Ts owing to positive terms of order γ2. If Ts < 0, the superconductivity appears,provided Tm > 0, i.e., when γ2/4κγ1 > |Ts| (see Table 1).

5. Temperature-pressure phase diagram

Although the structure of the FS phase is quite complicated, some of the results can beobtained in analytical form. A more detailed outline of the phase domains, for example, inT − P phase diagram, can be done by using suitable values of the material parameters in thefree energy (7): P0, Tf 0, Ts, κ, γ, and γ1. Here we present some of the analytical results for thephase transition lines and the multi-critical points. Typical shapes of phase diagrams deriveddirectly from Eq. (7) are given in Figs. 2–7. Figure 2 shows the phase diagram calculated fromEq. (7) for parameters, corresponding to the experimental data [13] for ZrZn2. Figures 3 and 4show the low-temperature and the high-pressure parts of the same phase diagram (see Sec. 7for details). Figures 5–7 show the phase diagram calculated for the experimental data [2, 4]of UGe2 (see Sec. 8). In ZrZn2, UGe2, as well as in UCoGe and UIr, critical pressure Pc exists,where both superconductivity and ferromagnetic orders vanish.

As in experiments, we find out from our calculation that in the vicinity of P0 ∼ Pc the FM-FSphase transition is of fist order, denoted by the solid line BC in Figs. 3, 4, 6, and 7. At lowerpressure the same phase transition is of second orderq shown by the dotted lines in the samefigures. The second order phase transition line TFS(P) separating the FM and FS phases isgiven by the solution of the equation

TFS(P) = Ts + γ1tFS(P) + γ[−tFS(P)]1/2, (17)

where tFS(P) = t(TFS , P) ≤ 0, γ = γ/κ, γ1 = γ1/κ, and 0 < P < PB; PB is the pressurecorresponding to the multi-critical point B, where the line TFS(P) terminates, as clearly shownin Figs. 4 and 7). Note, that Eq. (17) strictly coincides with the stability condition for the FMphase with respect to appearance of FS phase [20].

Additional information for the shape of this phase transition line can be obtained by thederivative ρ = ∂TFS(P)/∂P, namely,

ρ =ρs + γ1 − γ/2(−tFS)

1/2

1 − nTn−1FS

[γ1 − γ/2[(−tFS)1/2

] , (18)

where ρs = ∂Ts(P)/∂P. Note, that Eq. (18) is obtained from Eqs. (10) and (17).

The shape of the line TFS(P) can vary depending on the theory parameters (see, e.g., Figs.3and 6). For certain ratios of γ, γ1, and values of ρs, the curve TFS(P) exhibits a maximumTm = TFS(Pm), given by ρ(ρs, Tm, Pm) = 0. This maximum is clearly seen in Figs. 6 and 7. Tolocate the maximum we need to know ρs. We have already assumed Ts does not depend onP, as explained above, which from the physical point of view means that the function Ts(P) is



flat enough to allow the approximation Ts ≈ 0 without a substantial error in the results. Fromour choice of P-dependence of the free energy [Eq. (7)] parameters, it follow that ρs = 0.

Setting ρs = ρ = 0 in Eq. (18) we obtain

t(Tm, Pm) = − γ2

4γ21

, (19)

namely, the value tm(T, P) = t(Tm, Pm) at the maximum Tm(Pm) of the curve TFS(P).Substituting tm back in Eq. (17) we obtain Tm, and with its help we also obtain the pressurePm, both given in Table 1, respectively.

We want to draw the attention to a particular feature of the present theory that the coordinatesTm and Pm of the maximum (point max) at the curve TFS(P) as well as the results fromvarious calculations with the help of Eqs. (17) and (18) are expressed in terms of the reducedinteraction parameters γ and γ1. Thus, using certain experimental data for Tm, Pm, as wellas Eqs. (17) and (18) for TFS, Ts, and the derivative ρ at particular values of the pressure P, γand γ1 can be calculated without any additional information, for example, for the parameterκ. This property of the model (7) is quite useful in the practical work with the experimentaldata.

Figure 2. T − P diagram of ZrZn2 calculated for Ts = 0, Tf 0 = 28.5 K, P0 = 21 kbar, κ = 10,γ = 2γ1 ≈ 0.2, and n = 1. The dotted line represents the FM-FS transition and the dashed line stands forthe second order N-FM transition. The dotted line has a zero slope at P = 0. The low-temperature andhigh-pressure domains of the FS phase are seen more clearly in the following Figs. 3 and 4.

The conditions for existence of a maximum on the curve TFS(P) can be determined byrequiring Pm > 0, and Tm > 0 and using the respective formulae for these quantities, shown inTable 1. This max always occurs in systems where TFS(0) ≤ 0 and the low-pressure part of thecurve TFS(P) terminates at T = 0 for some non-negative critical pressure P0c (see Sec. 6).But the max may occur also for some sets of material parameters, when TFS(0) > 0 (seeFig. 3, where Pm = 0). All these shapes of the line TFS(P) are described by the model (7).Irrespectively of the particular shape, the curve TFS(P) given by Eq. (17) always terminates atthe tricritical point (labeled B), with coordinates (PB, TB) (see, e.g., Figs. 4 and 7).



Figure 3. Details of Fig. 2 with expanded temperature scale. The points A, B, C are located in thehigh-pressure part (P ∼ Pc ∼ 21 kbar). The max point is at P ≈ 0 kbar. The FS phase domain is shaded.The dotted line shows the second order FM-FS phase transition with Pm ≈ 0. The solid straight line BCshows the fist-order FM-FS transition for P > PB. The quite flat solid line AC shows the first order N-FStransition (the lines BC and AC are more clearly seen in Fig. 4. The dashed line stands for the secondorder N-FM transition.

At pressure P > PB the FM-FS phase transition is of first order up to the critical-end pointC. For PB < P < PC the FM-FS phase transition is given by the straight line BC (see, e.g.,Figs. 4 and 7). The lines of all three phase transitions, N-FM, N-FS, and FM-FS, terminateat point C. For P > PC the FM-FS phase transition occurs on a rather flat smooth line ofequilibrium transition of first order up to a second tricritical point A with PA ∼ P0 and TA ∼ 0.Finally, the third transition line terminating at the point C describes the second order phasetransition N-FM. The reduced temperatures TN and pressures PN , N = (A, B, C, max) at thethree multi-critical points (A, B, and C), and the maximum Tm(Pm) are given in Table 1. Notethat, for any set of material parameters, TA < TC < TB < Tm and Pm < PB < PC < PA.

There are other types of phase diagrams, resulting from model (7). For negative values ofthe generic superconducting temperature Ts, several other topologies of the T − P diagramcan be outlined. The results for the multicritical points, presented in Table 1, shows that,when Ts lowers below T = 0, TC also decreases, first to zero, and then to negative values.When TC = 0 the direct N-FS phase transition of first order disappears and point C becomesa very special zero-temperature multicritical point. As seen from Table 1, this happens forTs = −γ2Tf (0)/4κ(1 + γ1). The further decrease of Ts causes point C to fall below the zerotemperature and then the zero-temperature phase transition of first order near Pc splits intotwo zero-temperature phase transitions: a second order N-FM transition and a first orderFM-FS transition, provided TB still remains positive.

At lower Ts also point B falls below T = 0 and the FM-FS phase transition becomes entirely ofsecond order. For very extreme negative values of Ts, a very large pressure interval below Pcmay occur where the FM phase is stable up to T = 0. Then the line TFS(P) will exist only forrelatively small pressure values (P � Pc). This shape of the stability domain of the FS phaseis also possible in real systems.



Figure 4. High-pressure part of the phase diagram of ZrZn2, shown in Fig. 1. The thick solid lines ACand BC show the first-order transitions N-FS, and FM-FS, respectively. Other notations are explained inFigs. 2 and 3.

6. Quantum phase transitions

We have shown that the free energy (6) describes zero temperature phase transitions. Usually,the properties of these phase transitions essentially depend on the quantum fluctuations ofthe order parameters. For this reason the phase transitions at ultralow and zero temperatureare called quantum phase transitions [31, 32]. The time-dependent quantum fluctuations(correlations) which describe the intrinsic quantum dynamics of spin-triplet ferromagneticsuperconductors at ultralow temperatures are not included in our consideration but somebasic properties of the quantum phase transitions can be outlines within the classical limitdescribed by the free energy models (6) and (7). Let we briefly clarify this point.

The classical fluctuations are entirely included in the general GL functional (1)–(5) but thequantum fluctuations should be added in a further generalization of the theory. Generally,both classical (thermal) and quantum fluctuations are investigated by the method of therenormalization group (RG) [31], which is specially intended to treat the generalized action ofsystem, where the order parameter fields (ϕ and M) fluctuate in time t and space �x [31, 32].These effects, which are beyond the scope of the paper, lead either to a precise treatment ofthe narrow critical region in a very close vicinity of second order phase transition lines or toa fluctuation-driven change in the phase-transition order. But the thermal fluctuations andquantum correlation effects on the thermodynamics of a given system can be unambiguouslyestimated only after the results from counterpart simpler theory, where these phenomenaare not present, are known and, hence, the distinction in the thermodynamic propertiespredicted by the respective variants of the theory can be established. Here we show that thebasic low-temperature and ultralow-temperature properties of the spin-triplet ferromagneticsuperconductors, as given by the preceding experiments, are derived from the model (6)without any account of fluctuation phenomena and quantum correlations. The latter might beof use in a more detailed consideration of the close vicinity of quantum critical points in thephase diagrams of ferromagnetic spin-triplet superconductors. Here we show that the theorypredicts quantum critical phenomena only for quite particular physical conditions whereas



the low-temperature and zero-temperature phase transitions of first order are favored by bothsymmetry arguments and detailed thermodynamic analysis.

There is a number of experimental [9, 40] and theoretical [17, 41, 42] investigations of theproblem for quantum phase transitions in unconventional ferromagnetic superconductors,including the mentioned intermetallic compounds. Some of them are based on differenttheoretical schemes and do not refer to the model (6). Others, for example, those inRef. [41] reported results about the thermal and quantum fluctuations described by themodel (6) before the comprehensive knowledge for the results from the basic treatmentreported in the present investigation. In such cases one could not be sure about the correctinterpretation of the results from the RG and the possibilities for their application to particularzero-temperature phase transitions. Here we present basic results for the zero-temperaturephase transitions described by the model (6).

Figure 5. T − P diagram of UGe2 calculated taking Ts = 0, Tf 0 = 52 K, P0 = 1.6 GPa, κ = 4, γ = 0.0984,γ1 = 0.1678, and n = 1. The dotted line represents the FM-FS transition and the dashed line stands forthe N-FM transition. The low-temperature and high-pressure domains of the FS phase are seen moreclearly in the following Figs. 6 and 7.

The RG investigation [41] has demonstrated up to two loop order of the theory that thethermal fluctuations of the order parameter fields rescale the model (6) in a way whichcorresponds to first order phase transitions in magnetically anisotropic systems. This resultis important for the metallic compounds we consider here because in all of them magneticanisotropy is present. The uniaxial magnetic anisotropy in ZrZn2 is much weaker than inUGe2 but cannot be neglected when fluctuation effects are accounted for. Owing to theparticular symmetry of model (6), for the case of magnetic isotropy (Heisenberg symmetry),the RG study reveals an entirely different class of (classical) critical behavior. Besides, thedifferent spatial dimensions of the superconducting and magnetic quantum fluctuationsimply a lack of stable quantum critical behavior even when the system is completelymagnetically isotropic. The pointed arguments and preceding results lead to the reliableconclusion that the phase transitions, which have already been proven to be first order in thelowest-order approximation, where thermal and quantum fluctuations are neglected, will not



undergo a fluctuation-driven change in the phase transition order from first to second. Suchpicture is described below, in Sec. 8, and it corresponds to the behavior of real compounds.

Our results definitely show that the quantum phase transition near Pc is of first order. Thisis valid for the whole N-FS phase transition below the critical-end point C, as well as thestraight line BC. The simultaneous effect of thermal and quantum fluctuations do not changethe order of the N-FS transition, and it is quite unlikely to suppose that thermal fluctuationsof the superconductivity field ψ can ensure a fluctuation-driven change in the order ofthe FM-FS transition along the line BC. Usually, the fluctuations of ψ in low temperaturesuperconductors are small and slightly influence the phase transition in a very narrow criticalregion in the vicinity of the phase-transition point. This effect is very weak and can hardly beobserved in any experiment on low-temperature superconductors. Besides, the fluctuationsof the magnetic induction B always tend to a fluctuation-induced first-order phase transitionrather than to the opposite effect - the generation of magnetic fluctuations with infinitecorrelation length at the equilibrium phase-transition point and, hence, a second order phasetransition [31, 43]. Thus we can quire reliably conclude that the first-order phase transitionsat low-temperatures, represented by the lines BC and AC in vicinity of Pc do not change theirorder as a result of thermal and quantum fluctuation fluctuations.

Figure 6. Low-temperature part of the T − P phase diagram of UGe2, shown in Fig. 5. The points A, B,C are located in the high-pressure part (P ∼ Pc ∼ 1.6 GPa). The FS phase domain is shaded. The thicksolid lines AC and BC show the first-order transitions N-FS, and FM-FS, respectively. Other notations areexplained in Figs. 2 and 3.

Quantum critical behavior for continuous phase transitions in spin-triplet ferromagneticsuperconductors with magnetic anisotropy can therefore be observed at otherzero-temperature transitions, which may occur in these systems far from the criticalpressure Pc. This is possible when TFS(0) = 0 and the TFS(P) curve terminates at T = 0 atone or two quantum (zero-temperature) critical points: P0c < Pm - “lower critical pressure",and P′

0c > Pm – “upper critical pressure." In order to obtain these critical pressures one shouldsolve Eq. (17) with respect to P, provided TFS(P) = 0, Tm > 0 and Pm > 0, namely, whenthe continuous function TFS(P) exhibits a maximum. The critical pressure P′

0c is bounded



in the relatively narrow interval (Pm, PB) and can appear for some special sets of materialparameters (r, t, γ, γ1). In particular, as our calculations show, P′

0c do not exists for Ts ≥ 0.

7. Criteria for type I and type II spin-tripletferromagnetic superconductors

The analytical calculation of the critical pressures P0c and P′0c for the general case of Ts = 0

leads to quite complex conditions for appearance of the second critical field P′0c. The correct

treatment of the case Ts = 0 can be performed within the entire two-domain picture for thephase FS (see, also, Ref. [20]). The complete study of this case is beyond our aims but herewe will illustrate our arguments by investigation of the conditions, under which the criticalpressure Poc occurs in systems with Ts ≈ 0. Moreover, we will present the general result forP0c ≥ 0 and P′

0c ≥ 0 in systems where Ts = 0.

Figure 7. High-pressure part of the phase diagram of UGe2, shown in Fig. 4. Notations are explained inFigs. 2, 3, 5, and 6.

Setting TFS(P0c) = 0 in Eq. (17) we obtain the following quadratic equation,

γ1m20c − γm0c − Ts = 0, (20)

for the reduced magnetization,

m0c = [−t(0, Poc)]1/2 = (1 − P0c)

1/2 (21)

and, hence, for P0c. For Ts = 0, Eqs. (20) and (21) have two solutions with respect to P0c. Forsome sets of material parameters these solutions satisfy the physical requirements for P0c andP′

0c and can be identified with the critical pressures. The conditions for existence of P0c and P′0c

can be obtained either by analytical calculations or by numerical analysis for particular valuesof the material parameters.



For Ts = 0, the trivial solution P0c = 1 corresponds to P0c = P0 > PB and, hence, does notsatisfy the physical requirements. The second solution,

P0c = 1 − γ2

γ21

(22)

is positive forγ1γ

≥ 1 (23)

and, as shown below, it gives the location of the quantum critical point (T = 0, P0c < Pm). Atthis quantum critical point, the equilibrium magnetization m0c is given by m0c = γ/γ1 and istwice bigger that the magnetization mm = γ/2γ1 ([20]) at the maximum of the curve TFS(P).

To complete the analysis we must show that the solution (22) satisfies the condition P0c < Pm.By taking Pm from Table 1, we can show that solution (22) satisfies the condition P0c < Pm forn = 1, if

γ1 < 3κ, (24)

and for n = 2 (SFT case), when

γ < 2√

3κ. (25)

Finally, we determine the conditions under which the maximum Tm of the curve TFS(P) occursat non-negative pressures. For n = 1, we obtain that Pm ≥ 0 for n = 1, if

γ1γ

≥ 12

(1 +

γ1κ

)1/2, (26)

whereas for n = 2, the condition is

γ1γ

≥ 12

(1 +

γ2

4κ2

)1/2

. (27)

Obviously, the conditions (23)-(27) are compatible with one another. The condition (26) isweaker than the condition Eq. (23), provided the inequality (24) is satisfied. The same is validfor the condition (27) if the inequality (25) is valid. In Sec. 8 we will show that these theoreticalpredictions are confirmed by the experimental data.

Doing in the same way the analysis of Eq. (17), some results may easily obtained for Ts = 0.In this more general case the Eq. (17) has two nontrivial solutions, which yield two possiblevalues of the critical pressure

P0c(±) = 1 − γ2

4γ21

[1 ±

(1 +

4Tsκγ1

γ2

)1/2]2

. (28)

The relation P0c(−) ≥ P0c(+) is always true. Therefore, to have both P0c(±) ≥ 0, it is enough torequire P0c(+) ≥ 0. Having in mind that for the phase diagram shape, we study Tm > 0, and



according to the result for Tm in Table 1, this leads to the inequality Ts > −γ2/4κγ1. So, weobtain that P0c(+) ≥ 0 will exist, if

γ1γ

≥ 1 +κTs

γ, (29)

which generalizes the condition (23).

Now we can identify the pressure P0c(+) with the lower critical pressure P0c, and P0c(−) withthe upper critical pressure P′

0c. Therefore, for wide variations in the parameters, theory (6)describes a quantum critical point Poc, that exists, provided the condition (29) is satisfied.The quantum critical point (T = 0, P0c) exists in UGe2 and, perhaps, in other p-waveferromagnetic superconductors, for example, in UIr.

Our results predict the appearance of second critical pressure – the upper critical pressure P′oc

that exists under more restricted conditions and, hence, can be observed in more particularsystems, where Ts < 0. As mentioned in Sec. 5, for very extreme negative values of Ts, whenTB < 0, the upper critical pressure P′

0c > 0 occurs, whereas the lower critical pressure P0c >0 does not appear. Bue especially this situation should be investigated in a different way,namely, one should keep TFS(0) different from zero in Eq. (17), and consider a form of theFS phase domain in which the curve TFS(P) terminates at T = 0 for P′

0c > 0, irrespectiveof whether the maximum Tm exists or not. In such geometry of the FS phase domain, themaximum T(Pm) may exist only in quite unusual cases, if it exists at all.

Using criteria like (23) in Sec. 8.4 we classify these superconductors in two types: (i) type I,when the condition (23) is satisfied, and (ii) type II, when the same condition does not hold.As we show in Sec. 8.2, 8.3 and 8.4, the condition (23) is satisfied by UGe2 but the samecondition fails for ZrZn2. For this reason the phase diagrams of UGe2 and ZrZn2 exhibitqualitatively different shapes of the curves TFS(P). For UGe2 the line TFS(P) has a maximumat some pressure P > 0, whereas the line TFS(P), corresponding to ZrZn2, does not exhibitsuch maximum (see also Sec. 8).

The quantum and thermal fluctuation phenomena in the vicinities of the two critical pressuresP0c and P′

0c need a nonstandard RG treatment because they are related with the fluctuationbehavior of the superconducting field ψ far below the ferromagnetic phase transitions,where the magnetization M does not undergo significant fluctuations and can be considereduniform. The presence of uniform magnetization produces couplings of M and ψ which arenot present in previous RG studies and need a special analysis.

8. Application to metallic compounds

8.1. Theoretical outline of the phase diagram

In order to apply the above displayed theoretical calculations, following from free energy (7),for the outline of T − P diagram of any material, we need information about the values of P0,Tf 0, Ts, κ γ, and γ1. The temperature Tf 0 can be obtained directly from the experimentalphase diagrams. The pressure P0 is either identical or very close to the critical pressurePc, for which the N-FM phase transition line terminates at T ∼ 0. The temperature Tsof the generic superconducting transition is not available from the experiments because, asmentioned above, pure superconducting phase not coexisting with ferromagnetism has not



been observed. This can be considered as an indication that Ts is very small and does notproduce a measurable effect. So the generic superconducting temperature will be estimatedon the basis of the following arguments. For Tf (P) > Ts we must have Ts(P) = 0 at P ≥ Pc,where Tf (P) ≤ 0, and for 0 ≤ P ≤ P0, Ts < TC. Therefore for materials where TC is too smallto be observed experimentally, Ts can be ignored.

As far as the shape of FM-FS transition line is well described by Eq. (17), we will make use ofadditional data from available experimental phase diagrams for ferroelectric superconductors.For example, in ZrZn2 these are the observed values of TFS(0) and the slope ρ0 ≡[∂TFS(P)/∂P]0 = (Tf 0/P0)ρ0 at P = 0; see Eq. (17). For UGe2, where a maximum (Tm) isobserved on the phase-transition line, we can use the experimental values of Tm, Pm, and P0c.The interaction parameters γ and γ1 are derived using Eq. (17), and the expressions for Tm,Pm, and ρ0, see Table 1. The parameter κ is chosen by fitting the expression for the critical-endpoint TC.

8.2. ZrZn2

Experiments for ZrZn2 [13] gives the following values: Tf 0 = 28.5 K, TFS(0) = 0.29 K, P0 ∼Pc = 21 kbar. The curve TF(P) ∼ Tf (P) is almost a straight line, which directly indicates thatn = 1 is adequate in this case for the description of the P-dependence. The slope for TFS(P)at P = 0 is estimated from the condition that its magnitude should not exceed Tf 0/Pc ≈ 0.014as we have assumed that is straight one, so as a result we have −0.014 < ρ ≤ 0. This ignoresthe presence of a maximum. The available experimental data for ZrZn2 do not give clearindication whether a maximum at (Tm, Pm) exists. If such a maximum were at P = 0 wewould have ρ0 = 0, whereas a maximum with Tm ∼ TFS(0) and Pm � P0 provides us withan estimated range 0 ≤ ρ0 < 0.005. The choice ρ0 = 0 gives γ ≈ 0.02 and γ1 ≈ 0.01,but similar values hold for any |ρ0| ≤ 0.003. The multicritical points A and C cannot bedistinguished experimentally. Since the experimental accuracy [13] is less than ∼ 25 mK inthe high-P domain (P ∼ 20 − 21 kbar), we suppose that TC ∼ 10 mK, which corresponds toκ ∼ 10. We employed these parameters to calculate the T − P diagram using ρ0 = 0 and 0.003.The differences obtained in these two cases are negligible, with both phase diagrams being inexcellent agreement with experiment.

Phase diagram of ZrZn2 calculated directly from the free energy (7) for n = 1, the abovementioned values of Ts, P0, Tf 0, κ, and values of γ ≈ 0.2 and γ1 ≈ 0.1 which ensure ρ0 ≈ 0 isshown in Fig. 2. Note, that the experimental phase diagram [13] of ZrZn2 looks almost exactlyas the diagram in Fig. 2, which has been calculated directly from the model (7) without anyapproximations and simplifying assumptions. The phase diagram in Fig. 2 has the followingcoordinates of characteristic points: PA ∼ Pc = 21.42 kbar, PB = 20.79 kbar, PC = 20.98 kbar,TA = TF(Pc) = TFS(Pc) = 0 K, TB = 0.0495 K, TC = 0.0259 K, and TFS(0) = 0.285 K.

The low-T region is seen in more detail in Fig. 3, where the A, B, C points are shown andthe order of the FM-FS phase transition changes from second to first order around the criticalend-point C. The TFS(P) curve, shown by the dotted line in Fig. 3, has a maximum Tm = 0.290K at P = 0.18 kbar, which is slightly above TFS(0) = 0.285 K. The straight solid line BC inFig. 3 shows the first order FM-FS phase transition which occurs for PB < P < PC. The solidAC line shows the first order N-FS phase transition and the dashed line stands for the N-FMphase transition of second order.



Although the expanded temperature scale in Fig. 3, the difference [Tm − TFS(0)] = 5 mK ishard to see. To locate the point max exactly at P = 0 one must work with values of γ andγ1 of accuracy up to 10−4. So, the location of the max for parameters corresponding to ZrZn2is very sensitive to small variations of γ and γ1 around the values 0.2 and 0.1, respectively.Our initial idea was to present a diagram with Tm = TFS(0) = 0.29 K and ρ0 = 0, namely,max exactly located at P = 0, but the final phase diagram slightly departs from this picturebecause of the mentioned sensitivity of the result on the values of the interaction parametersγ and γ1. The theoretical phase diagram of ZrZn2 can be deduced in the same way for ρ0 =0.003 and this yields Tm = 0.301 K at Pm = 6.915 kbar for initial values of γ and γ1 whichdiffers from γ = 2γ1 = 0.2 only by numbers of order 10−3 − 10−4 [18]. This result confirmsthe mentioned sensitivity of the location of the maximum Tm towards slight variations ofthe material parameters. Experimental investigations of this low-temperature/low-pressureregion with higher accuracy may help in locating this maximum with better precision.

Fig. 4 shows the high-pressure part of the same phase diagram in more details. In this figurethe first order phase transitions (solid lines BC and AC) are clearly seen. In fact the line AC isquite flat but not straight as the line BC. The quite interesting topology of the phase diagramof ZrZn2 in the high-pressure domain (PB < P < PA) is not seen in the experimental phasediagram [13] because of the restricted accuracy of the experiment in this range of temperaturesand pressures.

These results account well for the main features of the experimental behavior [13], includingthe claimed change in the order of the FM-FS phase transition at relatively high P. Withinthe present model the N-FM transition is of second order up to PC ∼ Pc. Moreover, if theexperiments are reliable in their indication of a first order N-FM transition at much lower Pvalues, the theory can accommodate this by a change of sign of b f , leading to a new tricriticalpoint located at a distinct Ptr < PC on the N-FM transition line. Since TC > 0 a directN-FS phase transition of first order is predicted in accord with conclusions from de Haas–vanAlphen experiments [44] and some theoretical studies [40]. Such a transition may not occur inother cases where TC = 0. In SFT (n = 2) the diagram topology remains the same but pointsB and C are slightly shifted to higher P (typically by about 0.01 −−0.001 kbar).

8.3. UGe2

The experimental data for UGe2 indicate Tf 0 = 52 K, Pc = 1.6 GPa (≡ 16 kbar), Tm = 0.75 K,Pm ≈ 1.15 GPa, and P0c ≈ 1.05 GPa [2–5]. Using again the variant n = 1 for Tf (P) and theabove values for Tm and P0c we obtain γ ≈ 0.0984 and γ1 ≈ 0.1678. The temperature TC ∼ 0.1K corresponds to κ ∼ 4.

Using these initial parameters, together with Ts = 0, leads to the T − P diagram of UGE2shown in Fig. 5. We obtain TA = 0 K, PA = 1.723 GPa, TB = 0.481 K, PB = 1.563 GPa, TC =0.301 K, and PC = 1.591 GPa. Figs. 6 and 7 show the low-temperature and the high-pressureparts of this phase diagram, respectively. There is agreement with the main experimentalfindings, although Pm corresponding to the maximum (found at ∼ 1.44 GPa in Fig. 5) is about0.3 GPa higher than suggested experimentally [4, 5]. If the experimental plots are accuratein this respect, this difference may be attributable to the so-called (Tx) meta-magnetic phasetransition in UGe2, which is related to an abrupt change of the magnetization in the vicinityof Pm. Thus, one may suppose that the meta-magnetic effects, which are outside the scope ofour current model, significantly affect the shape of the TFS(P) curve by lowering Pm (along



with PB and PC). It is possible to achieve a lower Pm value (while leaving Tm unchanged), butthis has the undesirable effect of modifying Pc0 to a value that disagrees with experiment. InSFT (n = 2) the multi-critical points are located at slightly higher P (by about 0.01 GPa), as forZrZn2. Therefore, the results from the SFT theory are slightly worse than the results producedby the usual linear approximation (n = 1) for the parameter t.

8.4. Two types of ferromagnetic superconductors with spin-triplet electronpairing

The estimates for UGe2 imply γ1κ ≈ 1.9, so the condition for TFS(P) to have a maximumfound from Eq. (17) is satisfied. As we discussed for ZrZn2, the location of this maximum canbe hard to fix accurately in experiments. However, Pc0 can be more easily distinguished, as inthe UGe2 case. Then we have a well-established quantum (zero-temperature) phase transitionof second order, i.e., a quantum critical point at some critical pressure P0c ≥ 0. As shownin Sec. 6, under special conditions the quantum critical points could be two: at the lowercritical pressure P0c < Pm and the upper critical pressure P′

0c < Pm. This type of behaviorin systems with Ts = 0 (as UGe2) occurs when the criterion (23) is satisfied. Such systems(which we label as U-type) are essentially different from those such as ZrZn2 where γ1 < γand hence TFS(0) > 0. In this latter case (Zr-type compounds) a maximum Tm > 0 maysometimes occur, as discussed earlier. We note that the ratio γ/γ1 reflects a balance effectbetween the two ψ-M interactions. When the trigger interaction (typified by γ) prevails, theZr-type behavior is found where superconductivity exists at P = 0. The same ratio can beexpressed as γ0/δ0M0, which emphasizes that the ground state value of the magnetization atP = 0 is also relevant. Alternatively, one may refer to these two basic types of spin-tripletferromagnetic superconductors as "type I" (for example, for the "Zr-type compounds), and"type II" – for the U-type compounds.

As we see from this classification, the two types of spin-triplet ferromagnetic superconductorshave quite different phase diagram topologies although some fragments have commonfeatures. The same classification can include systems with Ts = 0 but in this case one shoulduse the more general criterion (29).

8.5. Other compounds

In URhGe, Tf (0) ∼ 9.5 K and TFS(0) = 0.25 K and, therefore, as in ZrZn2, here thespin-triplet superconductivity appears at ambient pressure deeply in the ferromagnetic phasedomain [6–8]. Although some similar structural and magnetic features are found in UGe2the results in Ref. [8] of measurements under high pressure show that, unlike the behavior ofZrZn2 and UGe2, the ferromagnetic phase transition temperature TF(P) ∼ Tf (P) has a slowlinear increase up to 140 kbar without any experimental indications that the N-FM transitionline may change its behavior at higher pressures and show a negative slope in direction oflow temperature up to a quantum critical point TF = 0 at some critical pressure Pc. Such abehavior of the generic ferromagnetic phase transition temperature cannot be explained byour initial assumption for the function Tf (P) which was intended to explain phase diagramswhere the ferromagnetic order is depressed by the pressure and vanishes at T = 0 at somecritical pressure Pc. The TFS(P) line of URhGe shows a clear monotonic negative slopeto T = 0 at pressures above 15 kbar and the extrapolation [8] of the experimental curveTFS(P) tends a quantum critical point TFS(P′

oc) = 0 at P0c ∼ 25 − 30 kbar. Within the



framework of the phenomenological theory (6, this T − P phase diagram can be explainedafter a modification on the Tf (P)-dependence is made, and by introducing a convenientnontrivial pressure dependence of the interaction parameter γ. Such modifications of thepresent theory are possible and follow from important physical requirements related with thebehavior of the f -band electrons in URhGe. Unlike UGe2, where the pressure increases thehybridization of the 5 f electrons with band states lading to a suppression of the spontaneousmagnetic moment M, in URhGe this effects is followed by a stronger effect of enhancementof the exchange coupling due to the same hybridization, and this effect leads to the slow butstable linear increase in the function TF(P)[8]. These effects should be taken into account inthe modeling the pressure dependence of the parameters of the theory (7) when applied toURhGe.

Another ambient pressure FS phase has been observed in experiments with UCoGe [9]. Herethe experimentally derived slopes of the functions TF(P) and TFS(P) at relatively smallpressures are opposite compared to those for URhGe and, hence, the T − P phase diagramof this compound can be treated within the present theoretical scheme without substantialmodifications.

Like in UGe2, the FS phase in UIr [12] is embedded in the high-pressure/low-temperaturepart of the ferromagnetic phase domain near the critical pressure Pc which means that UIr iscertainly a U-type compound. In UGe2 there is one metamagnetic phase transition betweentwo ferromagnetic phases (FM1 and FM2), in UIr there are three ferromagnetic phases andthe FS phase is located in the low-T/high-P domain of the third of them - the phase FM3.There are two metamagnetic-like phase transitions: FM1-FM2 transition which is followedby a drastic decrease of the spontaneous magnetization when the the lower-pressure phaseFM1 transforms to FM2, and a peak of the ac susceptibility but lack of observable jump of themagnetization at the second (higher pressure) “metamagnetic" phase transition from FM2 toFM3. Unlike the picture for UGe2, in UIr both transitions, FM1-FM2 and FM2-FM3 are farfrom the maximum Tm(Pm) so in this case one can hardly speculate that the max is producedby the nearby jump of magnetization. UIr seems to be a U-type spin-triplet ferromagneticsuperconductor.

9. Final remarks

Finally, even in its simplified form, this theory has been shown to be capable of accountingfor a wide variety of experimental behavior. A natural extension to the theory is to add a M6

term which provides a formalism to investigate possible metamagnetic phase transitions [45]and extend some first order phase transition lines. Another modification of this theory, withregard to applications to other compounds, is to include a P dependence for some of the otherGL parameters. The fluctuation and quantum correlation effects can be considered by therespective field-theoretical action of the system, where the order parameters ψ and M are notuniform but rather space and time dependent. The vortex (spatially non-uniform) phase dueto the spontaneous magnetization M is another phenomenon which can be investigated by ageneralization of the theory by considering nonuniform order parameter fields ψ and M (see,e.g., Ref. [28]). Note that such theoretical treatments are quite complex and require a numberof approximations. As already noted in this paper the magnetic fluctuations stimulate firstorder phase transitions for both finite and zero phase-transition temperatures.



Author details

Dimo I. UzunovCollective Phenomena Laboratory, G. Nadjakov Institute of Solid State Physics, Bulgarian Academy ofSciences, BG-1784 Sofia, Bulgaria.Pacs: 74.20.De, 74.25.Dw, 64.70.Tg

10. References

[1] D. Vollhardt and P. Wölfle, it The Superfluid Phases of Helium 3 (Taylor & Francis,London, 1990); D. I. Uzunov, in: Advances in Theoretical Physics, edited by E. Caianiello(World Scientific, Singapore, 1990), p. 96; M. Sigrist and K. Ueda, Rev. Mod. Phys. 63,239 (1991).

[2] S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche, R. K. W. Haselwimmer, M.J. Steiner,E. Pugh, I. R. Walker, S.R. Julian, P. Monthoux, G. G. Lonzarich, A. Huxley. I. Sheikin, D.Braithwaite, and J. Flouquet, Nature 406, 587 (2000).

[3] A. Huxley, I. Sheikin, E. Ressouche, N. Kernavanois, D. Braithwaite, R. Calemczuk, andJ. Flouquet, Phys. Rev. B63, 144519 (2001).

[4] N. Tateiwa, T. C. Kobayashi, K. Hanazono, A. Amaya, Y. Haga. R. Settai, and Y. Onuki,J. Phys. Condensed Matter 13, L17 (2001).

[5] A. Harada, S. Kawasaki, H. Mukuda, Y. Kitaoka, Y. Haga, E. Yamamoto, Y. Onuki, K. M.Itoh, E. E. Haller, and H. harima, Phys. Rev. B 75, 140502 (2007).

[6] D. Aoki, A. Huxley, E. Ressouche, D. Braithwaite, J. Flouquet, J-P.. Brison, E. Lhotel, andC. Paulsen, Nature 413, 613 (2001).

[7] F. Hardy, A. Huxley, Phys. Rev. Lett. 94, 247006 (2005).[8] F. Hardy, A. Huxley, J. Flouquet, B. Salce, G. Knebel, D. Braithwate, D. Aoki, M. Uhlarz,

and C. Pfleiderer, Physica B 359-361 1111 (2005).[9] N. T. Huy, A. Gasparini, D. E. de Nijs, Y. Huang, J. C. P. Klaasse, T. Gortenmulder, A. de

Visser, A. Hamann, T. Görlach, and H. v. Löhneysen, Phys. Rev. Lett. 99, 067006 (2007).[10] N. T. Huy, D. E. de Nijs, Y. K. Huang, and A. de Visser, Phys. Rev. Lett. 100, 077001

(2008).[11] T. Akazawa, H. Hidaka, H. Kotegawa, T. C. Kobayashi, T. Fujiwara, E. Yamamoto, Y.

Haga, R. Settai, and Y. Onuki, Physica B 359-361, 1138 (2005).[12] T. C. Kobayashi,S. Fukushima, H. Hidaka, H. Kotegawa, T. Akazawa, E. Yamamoto, Y.

Haga, R. Settai, and Y. Onuki, Physica B 378-361, 378 (2006).[13] C. Pfleiderer, M. Uhlatz, S. M. Hayden, R. Vollmer, H. v. Löhneysen, N. R. Berhoeft, and

G. G. Lonzarich, Nature 412, 58 (2001).[14] E. A. Yelland, S. J. C. Yates, O. Taylor, A. Griffiths, S. M. Hayden, and A. Carrington,

Phys. Rev. B 72, 184436 (2005).[15] E. A. Yelland, S. M. Hayden, S. J. C. Yates, C. Pfleiderer, M. Uhlarz, R. Vollmer, H. v

Löhneysen, N. R. Bernhoeft, R. P. Smith, S. S. Saxena, and N. Kimura, Phys. Rev. B72,214523 (2005).

[16] C. J. Bolesh and T. Giamarchi, Phys. Rev. Lett. 71, 024517 (2005); R. D. Duncan, C.Vaccarella, and C. A. S. de Melo, Phys. Rev. B 64, 172503 (2001).

[17] A. H. Nevidomskyy, Phys. Rev. Lett. 94, 097003 (2005).[18] M. G. Cottam, D. V. Shopova and D. I. Uzunov, Phys. Lett. A 373, 152 (2008).



[19] D. V. Shopova and D. I. Uzunov, Phys. Rev. B 79, 064501 (2009).[20] D. V. Shopova and D. I. Uzunov, Phys. Rev. 72, 024531 (2005); Phys. Lett. A 313, 139

(2003).[21] D. V. Shopova and D. I. Uzunov, in: Progress in Ferromagnetism Research, ed. by V. N.

Murray (Nova Science Publishers, New York, 2006), p. 223; D. V. Shopova and D. I.Uzunov, J. Phys. Studies , 4, 426 (2003) 426; D. V. Shopova and D. I. Uzunov, Compt.Rend Acad. Bulg. Sci. 56, 35 (2003) 35; D. V. Shopova, T. E. Tsvetkov, and D. I. Uzunov,Cond. Matter Phys. 8, 181 (2005) 181; D. V. Shopova, and D. I. Uzunov, Bulg. J. of Phys.32, 81 (2005).

[22] E. K. Dahl and A. Sudbø, Phys. Rev. B 75, 1444504 (2007).[23] K. Machida and T. Ohmi, Phys. Rev. Lett. 86, 850 (2001).[24] M. B. Walker and K. V. Samokhin, Phys. Rev. Lett. 88, 207001 (2002); K. V. Samokhin and

M. B. Walker, Phys. Rev. B 66, 024512 (2002); Phys. Rev. B 66, 174501 (2002).[25] J. Linder, A. Sudbø, Phys. Rev. B 76, 054511 (2007); J. Linder, I. B. Sperstad, A. H.

Nevidomskyy, M. Cuoco, and A. Sodbø, Phys. Rev. 77, 184511 (2008); J. Linder, T.Yokoyama, and A. Sudbø, Phys. Rev. B 78, 064520 (2008); J. Linder, A. H. Nevidomskyy,A. Sudbø, Phys. Rev. B 78, 172502 (2008).

[26] R. A. Cowley, Adv. Phys. 29, 1 (1980); J-C. Tolédano and P. Tolédano, The Landau Theoryof Phase Transitions (World Scientific, Singapore, 1987).

[27] S. V. Vonsovsky, Yu. A. Izyumov, and E. Z. Kurmaev, Superconductivity of TransitionMetals (Springer Verlag, Berlin, 1982).

[28] L. N. Bulaevskii, A. I Buzdin, M. L. Kulic, and S. V. Panyukov, Adv. Phys. 34, 175 (1985);Sov. Phys. Uspekhi, 27, 927 (1984). A. I. Buzdin and L. N. Bulaevskii, Sov. Phys. Uspekhi29, 412 (1986).

[29] E. I. Blount and C. M. Varma, Phys. Rev. Lett. 42, 1079 (1979).[30] K. K. Murata and S. Doniach, Phys. Rev. Lett. 29, 285 (1972); G. G. Lonzarich and L.

Taillefer, J. Phys. C: Solid State Phys. 18, 4339 (1985); T. Moriya, J. Phys. Soc. Japan 55,357 (1986); H. Yamada, Phys. Rev. B 47, 11211 (1993).

[31] D. I. Uzunov, Theory of Critical Phenomena, Second Edition (World Scientific, Singapore,2010).

[32] D. V. Shopova and D. I. Uzunov, Phys. Rep. C 379, 1 (2003).[33] A. A. Abrikosov, Zh. Eksp. Teor. Fiz. 32, 1442 (1957) [Sov. Phys. JETP 5q 1174 (1957)].[34] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, II Part (Pergamon Press, London,

1980) [Landau-Lifshitz Course in Theoretical Physics, Vol. IX].[35] H. Belich, O. D. Rodriguez Salmon, D. V. Shopova and D. I. Uzunov, Phys. Lett. A 374,

4161 (2010); H. Belich and D. I. Uzunov, Bulg. J. Phys. 39, 27 (2012).[36] E. P. Wohlfarth, J. Appl. Phys. 39, 1061 (1968); Physica B&C 91B, 305 (1977).[37] P. Misra, Heavy-Fermion Systems, (Elsevier, Amsterdam, 2008).[38] K. G. Sandeman, G. G. Lonzarich, and A. J. Schofield, Phys. Rev. Lett. 90, 167005 (2003).[39] T. F. Smith, J. A. Mydosh, and E. P. Wohlfarth, Phys. rev. Lett. 27, 1732 (1971); G. Oomi,

T. Kagayama, K. Nishimura, S. W. Yun, and Y. Onuki, Physica B 206, 515 (1995).[40] M. Uhlarz, C. Pfleiderer, and S. M. Hayden, Phys. Rev. Lett. 93, 256404 (2004).[41] D. I. Uzunov, Phys. Rev. B74, 134514 (2006); Europhys. Lett. 77, 20008 (2007).[42] D. Belitz, T. R. Kirkpatrick, J. Rollbühler, Phys. Rev. Lett. 94, 247205 (2005); G. A.

Gehring, Europhys. Lett. 82, 60004 (2008).



[43] B. I. Halperin, T. C. Lubensky, and S. K. Ma, Phys. Rev. Lett. 32, 292 (1974); J-H. Chen,T. C. Lubensky, and D. R. Nelson, Phys. Rev. B17, 4274 (1978).

[44] N. Kimura et al., Phys. Rev. Lett. 92 , 197002 (2004).[45] A. Huxley, I. Sheikin, and D. Braithwaite, Physica B 284-288, 1277 (2000).


theory of ferromagnetic unconventional superconductors ......theory of ferromagnetic unconventional...

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